Description Of Electron Transfer Research Articles

On the use of energy level diagrams for semiconducting ionic electrodes

General treatment of charge transfer at electrochemical interfaces is based on charge carrier concentration and activation enthalpy of the transfer step. For the description of electron transfer at semiconductor electrodes, energy-level diagrams have proved useful, giving a detailed picture of the nature of the interface. In the case of semiconducting ionic electrodes, however, energy-level diagrams are currently not being utilized. The reason is insufficient knowledge regarding ion energy levels and nature of double layer. This contribution briefly reviews the fundamentals of charge transfer at semiconducting electrodes and discusses the establishment of energy-level diagrams for semiconducting ionic electrodes.

Hybrid density functional theory meets quasiparticle calculations: A consistent electronic structure approach

We propose a scheme to obtain a system-dependent fraction of exact exchange ($\ensuremath{\alpha}$) within the framework of hybrid density functional theory (DFT) that is consistent with the ${G}_{0}{W}_{0}$ approach, where ${G}_{0}$ is the noninteracting Green function of the system and ${W}_{0}$ the screened Coulomb interaction. We exploit the formally exact condition of exact DFT that the energy of the highest occupied molecular orbital corresponds to the ionization potential of a finite system. We identify the optimal $\ensuremath{\alpha}$ value for which this statement is obeyed as closely as possible and thereby remove the starting point dependence from the ${G}_{0}{W}_{0}$ method. This combined approach is essential for describing electron transfer (as exemplified by the TTF/TCNQ dimer) and yields the vertical ionization potentials of the G2 benchmark set with a mean absolute percentage error of only $\ensuremath{\approx}$3%.

Polaron formation: Ehrenfest dynamics vs. exact results

We use a one-dimensional tight binding model with an impurity site characterized by electron-vibration coupling, to describe electron transfer and localization at zero temperature, aiming to examine the process of polaron formation in this system. In particular we focus on comparing a semiclassical approach that describes nuclear motion in this many vibronic-states system on the Ehrenfest dynamics level to a numerically exact fully quantum calculation based on the Bonca-Trugman method [J. Bonča and S. A. Trugman, Phys. Rev. Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear subspace is implemented in equivalent approximate ways: In the Ehrenfest calculation the uncoupled (to the electronic subsystem) motion of the classical (harmonic) oscillator is simply damped as would be implied by coupling to a Markovian zero temperature bath. In the quantum calculation, thermal relaxation is implemented by augmenting the Liouville equation for the oscillator density matrix with kinetic terms that account for the same relaxation. In both cases we calculate the probability to trap the electron by forming a polaron and the probability that it escapes to infinity. Comparing these calculations, we find that while both result in similar long time yields for these processes, the Ehrenfest-dynamics based calculation fails to account for the correct time scale for the polaron formation. This failure results, as usual, from the fact that at the early stage of polaron formation the classical nuclear dynamics takes place on an unphysical average potential surface that reflects the distributed electronic population in the system, while the quantum calculation accounts fully for correlations between the electronic and vibrational subsystems.

Structural and dynamic aspects of electron transfer in proteins — highly organized natural nanostructures

The review briefly outlines theoretical models developed in 1990s to describe electron transfer reactions (ETR) in proteins, as well as different variants of improvements in these models proposed by the present authors to describe ETR in reaction centers (RC) of photosynthetic bacteria with consideration of their molecular dynamics in a wide temperature range. Experimental data on electron transfer from reduced proximal heme c-559 of cytochrome to bacteriochlorophyll dimer radical cation P+ in RC from two types of bacteria, viz., native and mutant RC from Rps. viridis and native RC from Rps. sulfoviridis were analyzed within the framework of the models which take into account the quantum and classical (including diffusive) degrees of freedom responsible for reorganization of the protein globule.

Analysis of microsecond relaxation dynamics of proteins and viscous media by recording relaxation shifts of phosphorescence spectra

We studied the low-temperature dynamics of the Stokes shift of instantaneous phosphorescence spectra of eosin and eosin maleimide covalently bound to hemoglobin in a 66% v/v aqueous solution of glycerol under conditions of pulsed excitation. The kinetics of the Stokes shifts are described using the Cole-Davidson distribution function. From the experimental data, we obtain the parameters τ0 and β, which describe the Cole-Davidson distribution. Values of τ0 and β agree well with data obtained by other methods, and these parameters can be used to describe electron transfer reactions in polar solutions and proteins.

Metal-Mediated Electrochemical Oxidation of DNA-Wrapped Carbon Nanotubes

As part of the ongoing effort to describe electron transfer reactions of carbon nanotubes (CNTs), we studied the mediated electrochemical oxidation of CNTs solubilized by wrapping with a T(60) deoxyribooligonucleotide. Cyclic voltammetry revealed that the oxidation of this CNT-DNA material by electrogenerated ML(3)(3+) mediators completes a catalytic cycle that increases the oxidative current compared to that obtained by voltammetry of the mediator alone (M = Fe(III), Ru(III), or Os(III); L = 2,2'-bipyridine or 4,4'-dimethyl-2,2'-bipyridine). We observed a greater increase in current at higher nanotube concentration, slower experimental scan rate, and higher mediator redox potential (E(0)'). Using computer simulation, these observations were shown to be consistent with CNT oxidation involving the removal of multiple electrons from each CNT-DNA moiety (the T(60) oligonucleotide was chosen because it is not oxidized by any of the mediators). The data are well-described by a simulation model based on the classical catalytic mechanism (EC') with the following embellishment: three populations of CNT-DNA redox-active sites with different E(0)' and therefore different oxidation rates are employed to represent the varying redox potentials of different valence band electrons within one CNT chiral type and within the distribution of CNT types present in our sample. This modeling suggests the number of CNT-DNA sites available for oxidation increases with the E(0)' of the mediator. This result can be qualitatively interpreted in terms of CNT band theory.

Multistate electron transfer dynamics in the condensed phase: Exact calculations from the reduced hierarchy equations of motion approach

Multiple displaced oscillators coupled to an Ohmic heat bath are used to describe electron transfer (ET) in a dissipative environment. By performing a canonical transformation, the model is reduced to a multilevel system coupled to a heat bath with the Brownian spectral distribution. A reduced hierarchy equations of motion approach is introduced for numerically rigorous simulation of the dynamics of the three-level system with various oscillator configurations, for different nonadiabatic coupling strengths and damping rates, and at different temperatures. The time evolution of the reduced density matrix elements illustrates the interplay of coherences between the electronic and vibrational states. The ET reaction rates, defined as a flux-flux correlation function, are calculated using the linear response of the system to an external perturbation as a function of activation energy. The results exhibit an asymmetric inverted parabolic profile in a small activation regime due to the presence of the intermediate state between the reactant and product states and a slowly decaying profile in a large activation energy regime, which arises from the quantum coherent transitions.

Dynamics of electron transfer reactions in the presence of mode mixing: Comparison of a generalized master equation approach with the numerically exact simulation

A generalized master equation approach is developed to describe electron transfer (ET) dynamics in the presence of mode mixing. Results from this approximate approach are compared to the numerically exact simulations using the multilayer multiconfiguration time-dependent Hartree theory. The generalized master equation approach is found to work well for nonadiabatic resonant ET. Depending on the specific situation, it is found that the introduction of mode mixing may either increase or decrease the ET time scale. The master equation fails in the adiabatic ET regime, where the introduction of mode mixing may lead to electron trapping. From both the approximate theory and the numerically exact simulation it is shown how neglecting mode mixing in practical calculations may lead to inaccurate predictions of the ET dynamics.

Effectiveness of Perturbation Theory Approaches for Computing Non-Condon Electron Transfer Dynamics in Condensed Phases

A description of electron transfer in condensed-phase media requires models that adequately describe the coupling of the electronic degrees of freedom to the surrounding nuclear coordinates. The spin-boson model has been the canonical model used to understand quantum dynamic processes in condensed-phase media over the last 25 years. Inherent in the standard model of a two-state quantum system coupled to a bosonic bath is the assumption that the Condon approximation is valid. In this context, the Condon approximation assumes that the bath configurations (coordinates) have no effect on the nonadiabatic coupling matrix element. While this is a useful model for electron transfer in small molecular systems, the validity of this approximation is less likely when large-scale motions of solvent molecules are strongly coupled to the electron transfer event, e.g., in molecular clamps and long-range electron transfer in biopolymers. In the present paper a general model for two-state electron transfer which allows for system-bath coupling in both the diagonal and off-diagonal (nonadiabatic) terms is studied. Time-dependent perturbation theory for this Hamiltonian is developed using a small polaron transformation. As noted in several recent studies, in a certain regime of parameter space, the relevant Hamiltonian admits an exact solution, termed the exactly solvable non-Condon Hamiltonian (or NCE). This limit, for which exact solutions are available, is used to benchmark the short- and long-time accuracy of various perturbative approaches. The validated perturbation equations are subsequently used to explore the role of non-Condon effects on electron transfer by systematically increasing the strength of the non-Condon coupling term from zero (i.e., the canonical spin-boson model) to the value that pertains to the exactly solvable non-Condon model (where non-Condon effects are significant).

The initial and final states of electron and energy transfer processes: Diabatization as motivated by system-solvent interactions

For a system which undergoes electron or energy transfer in a polar solvent, we define the diabatic states to be the initial and final states of the system, before and after the nonequilibrium transfer process. We consider two models for the system-solvent interactions: A solvent which is linearly polarized in space and a solvent which responds linearly to the system. From these models, we derive two new schemes for obtaining diabatic states from ab initio calculations of the isolated system in the absence of solvent. These algorithms resemble standard approaches for orbital localization, namely, the Boys and Edmiston-Ruedenberg (ER) formalisms. We show that Boys localization is appropriate for describing electron transfer [Subotnik et al., J. Chem. Phys. 129, 244101 (2008)] while ER describes both electron and energy transfer. Neither the Boys nor the ER methods require definitions of donor or acceptor fragments and both are computationally inexpensive. We investigate one chemical example, the case of oligomethylphenyl-3, and we provide attachment/detachment plots whereby the ER diabatic states are seen to have localized electron-hole pairs.

Electron transfer dynamics: Zusman equation versus exact theory

The Zusman equation has been widely used to study the effect of solvent dynamics on electron transfer reactions. However, application of this equation is limited by the classical treatment of the nuclear degrees of freedom. In this paper, we revisit the Zusman equation in the framework of the exact hierarchical equations of motion formalism, and show that a high temperature approximation of the hierarchical theory is equivalent to the Zusman equation in describing electron transfer dynamics. Thus the exact hierarchical formalism naturally extends the Zusman equation to include quantum nuclear dynamics at low temperatures. This new finding has also inspired us to rescale the original hierarchical equations and incorporate a filtering algorithm to efficiently propagate the hierarchical equations. Numerical exact results are also presented for the electron transfer reaction dynamics and rate constant calculations.

Crystal and Molecular Structure of Manganese(II) Lapacholate, a Novel Polymeric Species Undergoing Temperature-Reversible Metal to Ligand Electron Transfer

Lapachol (2-hydroxy-3-(3-methyl-2-butenyl)-1,4-naphtoquinone) (HLap, C(15)H(14)O(3)) reacts with Mn(2+) producing a novel polymeric complex with formula: [Mn(Lap)(2)](n). Two ligands chelate the metal through their ortho oxygen (O1, O2) moiety while two para oxygens, from other Lap ligands, complete the octahedral coordination sphere. Thus far, all reported Lap metal complexes are mononuclear, lack the metal-trans-quinonic (para) oxygen binding and have Lap as a bidentate ligand. Synthesis, X-ray diffraction, IR, and UV-visible spectroscopic properties, thermogravimetric analysis, and differential thermal analysis of this complex are reported along with a density functional theory study describing electron transfer from the Mn to the Lap ligand at low temperature. X-ray structure determinations at 125, 197, and 300 K describe the progressive trend of a Mn contribution to the Mn-O1 bond length as a function of T. The Mn-O1 bond distance increases with temperature and may be therefore associated with a semiquinonate action at low T by the carbonyl O1 donor (and corresponding to Mn(III)). It transforms to a more classical coordinative bond at room T and stabilizes a Mn(II) species; this is a reversible phenomenon involving Mn(II)-Mn(III) oxidation states.

How does reorganization energy change upon protein unfolding? Monitoring the structural perturbations in the heme cavity of cytochrome c

In several classes of proteins the redox center provides an additional intrinsic biophysical probe that could be used to study the protein structure and function. In present report reorganization energy ( λ, as a parameter describing electron transfer properties) was used to study the protein structural changes around the heme prosthetic group in cytochrome c (cyt c). We attempted to monitor the value of this parameter upon the unfolding process of cyt c by urea, during which it was increased sigmoidally from about 0.52 to 0.82 eV for native and unfold protein, respectively. Results indicate that by structural changes in the heme site, λ provides a complementary tool for following the unfolding process. Assuming a reversible two-state model for cyt c unfolding, Δ G H 2O , C m and m values were determined to be 8.32 ± 0.7 kcal mol − 1 , 1.53 ±0.19 kcalmol − 1 M − 1 and 5.03 M, respectively.

Fe(II) hydrolysis in aqueous solution: A DFT study

Species arising from Fe(II) hydrolysis in aqueous solution have been investigated using density-functional methods (DFT). The different tautomers and multiplicities of each species have been calculated. The solvation energy has been estimated using the UAHF–PCM method. The hydrolysis free energies have been estimated and compared with the available experimental data. The different hydrolysis species have distinct geometries and electronic structures. The estimated ionization potential of the hydrolyzed species is linearly dependent to the number of hydroxyls present in the complex. The estimated Fe(II)/Fe(III) oxidation potential is in good agreement with previously published results about 0.29 V larger than the experimental value. The results highlight the importance of the chemical speciation in describing electron transfer processes at a molecular level. The PBE/TZVP/UAHF–PCM method has been found to describe correctly the hydrolysis free energies of Fe(II) with an average error about 5 kcal mol −1 from the experimental values.

Path-integral Monte Carlo simulations for electronic dynamics on molecular chains. II. Transport across impurities

Electron transfer (ET) across molecular chains including an impurity is studied based on a recently improved real-time path-integral Monte Carlo (PIMC) approach [L. Mühlbacher, J. Ankerhold, and C. Escher, J. Chem. Phys. 121 12696 (2004)]. The reduced electronic dynamics is studied for various bridge lengths and defect site energies. By determining intersite hopping rates from PIMC simulations up to moderate times, the relaxation process in the extreme long-time limit is captured within a sequential transfer model. The total transfer rate is extracted and shown to be enhanced for certain defect site energies. Super-exchange turns out to be relevant for extreme gap energies only and then gives rise to different dynamical signatures for high- and low-lying defects. Further, it is revealed that the entire bridge compound approaches a steady state on a much shorter time scale than that related to the total transfer. This allows for a simplified description of ET along donor-bridge-acceptor systems in the long-time range.

Combining molecular dynamics and ab initio quantum-chemistry to describe electron transfer reactions in electrochemical environments.

A theoretical model is presented aimed to provide a detailed microscopic description of the electron transfer reaction in an electrochemical environment. The present approach is based on the well-known two state model extended by the novelty that the energy of the two states involved in the electron transfer reaction is computed quantum mechanically as a function of the solvent coordinate, as defined in the Marcus theory, and of the intensity of an external electric field. The solvent conformations defining the reaction coordinate are obtained from classical molecular dynamics and then transferred to the quantum mechanical model. The overall approach has been applied to the electron transfer between a chloride anion and a single crystal Cu(100) electrode. It is found that the solvent exerts a strong influence on the equilibrium geometry of the halide and hence on the relative energy of the two states involved in the electron transfer reaction. Finally, both solvent fluctuations and external field facilitate the electron transfer although solvent effects have a stronger influence.

Low temperature extension of the generalized Zusman phase space equations for electron transfer.

In a previous paper [J. Chem. Phys. 119, 11864 (2003)], we derived a set of two coupled equations which describe electron transfer in the presence of dissipation at high temperature. Employing the low temperature extension of the Fokker-Planck operator, suggested by Haake and Reibold [Phys. Rev. A 32, 2462 (1985)] and Ankerhold [Europhys. Lett. 61, 301 (2003)], we show that one may extend the generalized Zusman equations in a similar manner to low temperature. Numerical simulation shows that addition of the temperature-dependent term which couples the coordinate and momentum causes an increase in the electron transfer rate as compared to the rate obtained from the previous high temperature equations. The increase in the rate comes from the increase in the equilibrium variances of the coordinate and momentum. The low temperature quantum theory allows for higher energy portions of phase space to contribute to the electron transfer rate where the rate is higher thus enhancing the overall rate.

A Nonlinear Dynamical Model for Ultrafast Catalytic Transfer of Electrons at Zero Temperature

The complex amplitudes of the electronic wavefunctions on different sites are used as Kramers variables for describing Electron Transfer. The strong coupling of the electronic charge to the many nuclei, ions, dipoles, etc, of the environment, is modeled as a thermal bath better considered classically. After elimination of the bath variables, the electron dynamics is described by a discrete nonlinear Schrödinger equation with norm preserving dissipative terms and Langevin random noises (at finite temperature). The standard Marcus results are recovered far from the inversion point, where atomic thermal fluctuations adiabatically induce the electron transfer. Close to the inversion point, in the non-adiabatic regime, electron transfer may become ultrafast (and selective) at low temperature essentially because of the nonlinearities, when these are appropriately tuned. We demonstrate and illustrate numerically that a weak coupling of the donor site with an extra appropriately tuned (catalytic) site, can trigger an ultrafast electron transfer to the acceptor site at zero degree Kelvin, while in the absence of this catalytic site no transfer would occur at all (the new concept of Targeted Transfer initially developed for discrete breathers is applied to polarons in our theory). Among other applications, this theory should be relevant for describing the ultrafast electron transfer observed in the photosynthetic reaction centers of living cells.

Primary charge separation in the bacterial reaction center: Validity of incoherent sequential model

A description of electron transfer (ET) by the incoherent sequential model was employed to elucidate the unidirectionality of the primary charge separation process in bacterial reaction centers (RC). The model assumes that the vibrational relaxation of the medium modes is sufficiently fast and that the system relaxes to thermal equilibrium after each ET step. ET was investigated for 5-sites (molecules) arranged in two branches. Beginning at molecule 1, ET can proceed in two directions with each branch composed of two molecules. Analysis shows that the model can successfully explain the asymmetry of primary electron transfer both in the wild type and several mutants of Rb capsulatus RC. In these cases the dependence of ET asymmetry on temperature was also evaluated. It was shown that in order to obtain the correct temperature dependence of ET asymmetry in the mutants, the superexchange mechanism operating in parallel with the sequential process must be used.

How Does the Solvent Control Electron Transfer? Experimental and Theoretical Studies of the Simplest Charge Transfer Reaction

The standard theoretical description used to describe electron transfer is Marcus theory, which maps the polarization of the solvent surrounding the reactants onto a reaction coordinate, q. The questions we address in this paper are: How many and what types of solvent degrees of freedom constitute q? Is it appropriate to treat the solvent as a dielectric continuum? Our approach to answer these questions relies on the study of the simplest possible charge transfer systems: we choose atomic systems that have only electronic degrees of freedom so that any spectroscopic changes that occur during the course of the reaction directly reflect the motions of the surrounding solvent. Our methods for characterizing these systems consist of both molecular dynamics (MD) simulations and femtosecond pump−probe experiments. Using MD, we find that even though solvent rotational motions appear to dominate the electronic relaxation when only the solute's charge changes, the slow translational motions of the few closest so...